THE THEOREM OF HONDA AND TATE

THE THEOREM OF HONDA AND TATE KIRSTEN EISENTRÄGER 1. Motivation Consider the following theorem of Tate (proved in [Mum70, Theorems 2 3, Appendix I]): Theorem 1.1. Let A and B be abelian varieties over a finite field k of size q, and let f A, f B Z[T ] be the characteristic polynomials of their q-frobenius endomorphisms, so f A and f B have degrees 2 dima and 2 dimb with constant terms q dima and q dimb respectively. The following statements are equivalent: (1) B is k-isogenous to an abelian subvariety of A. (2) f B f A in Q[T ]. In particular, A is k-isogenous to B if and only if f A = f B. Moreover, A is k-simple if and only if f A is a power of an irreducible polynomial in Q[T ]. Recall that the roots of f A in C are Weil q-integers : algebraic integers whose images in C all have absolute value q 1/2. By Theorem 1.1 to describe the isogeny classes of abelian varieties over finite fields we would like to know for which Weil q-integers π in C we can find a k- simple abelian variety A/k such that π is a root of f A. We will show that for each π, we can find a k-simple abelian variety A and an embedding Q[π] End 0 k(a) such that π is the q-frobenius endomorphism of A. We will follow the discussion in [Tat68] and [MW71]. Brian Conrad heavily revised Section 8 to simplify the computation of the invariant inv v (E) for v p and he provided the proofs for Theorem 8.3 and the results in the appendix. 2. Setup Let k be a finite field with q = p a elements. We will work in the category M(k) of abelian varieties up to isogeny over k: M(k) is the category whose objects are abelian varieties over k with the set of morphisms between two abelian varieties A and B given by Hom k (A, B) Q. Two abelian varieties are isomorphic in M(k) if 1

2 KIRSTEN EISENTRÄGER and only if they are isogenous over k. The category M(k) is a Q-linear category. Our goal is to describe the category M(k). Since every object in M(k) is a direct sum of a finite number of simple objects, it suffices to list the isomorphism classes of simple objects and, for each class, the endomorphism algebra. The theorems of Honda and Tate allow us to do this, as we shall see. Notation: For an abelian variety A over k, π A will denote the q-frobenius endomorphism. For abelian varieties A and B we use Hom 0 k(a, B) to denote Hom k (A, B) Q. Let A be an abelian variety over k, so π A commutes with all endomorphism of A and hence lies in the center of End 0 k(a). If A is simple, then End 0 k(a) is a division algebra. Therefore, when A is simple the subring Q[π] is a number field. An isogeny A B of simple abelian varieties over k defines an isomorphism End 0 k(a) End 0 k(b) carrying π A into π B and hence mapping Q[π A ] isomorphically onto Q[π B ]. We also know that π A is an algebraic integer whose absolute value is q 1/2 under every embedding into C. This motivates the following definition. Definition 2.1. A Weil q-integer in a field of characteristic 0 is an algebraic integer π such that for every embedding σ : Q[π] C, σπ = q 1/2. Let W (q) be the set of Weil q-integers in C. Definition 2.2. Two Weil q-integers π and π are conjugate, π π, if there exists an isomorphism of Q[π] Q[π ] carrying π into π. Equivalently, π and π have the same minimal polynomial over Q. Let A be a simple abelian variety defined over k. We will describe the division algebra End 0 k(a) in terms of π A. To do this, we first need to review the classification of division algebras over number fields. 3. Review: division algebras over global fields Definition 3.1. A central simple algebra over a field F is an F -algebra R such that (1) R is finite dimensional over F. (2) F is the center of R. (3) R is a simple ring. If R is also a division algebra, we call it a central division algebra over F. Notation: We will denote the r r matrix algebra over a field F by M r (F ).

THE THEOREM OF HONDA AND TATE 3 Lemma 3.2. If R and S are central simple algebras over F, then so is R F S. For any field F /F, R F F is a central simple algebra over F. Proof. For the first part see [Rei03, (7.6) Theorem], for the second statement see [Rei03, (7.8) Corollary]. Proposition 3.3 (Wedderburn s Theorem). Every central simple algebra over F is isomorphic to End E (E r ) = M r (E) for some r 1 and some central division algebra E over F. Moreover, r is uniquely determined by R, and E is uniquely determined up to F -isomorphism. Proof. See [Rei03, p. 91]. Definition 3.4 (Brauer group). Let A and B be finite dimensional central simple algebras over a field F. We say that A and B are similar, A B, if A F M n (F ) = M n (A) is F -isomorphic to B F M m (F ) = M m (B) for some m, n 1. Define the Brauer group of F, Br(F ), to be the set of similarity classes of central simple algebras over F, and write [A] for the similarity class of A. For classes [A] and [B], define [A][B] := [A F B]. This is well defined and makes Br(F ) into an abelian group with [A] 1 = A opp, where A opp is the opposite algebra. By Proposition 3.3, each element in Br(F ) is represented (up to isomorphism) by a unique central division algebra over F. We now need to clear up any possible confusion concerning sign conventions for local invariants of Brauer classes for non-archimedean local fields. Upon choosing a separable closure F s of an arbitrary field F, there are two natural procedures to define a functorial isomorphism Br(F ) H 2 (F s /F, F s ): a conceptual method via non-abelian cohomology as in [Ser79, Ch. X, 5] and an explicit method via crossed-product algebras. By [Ser79, Ch. X, 5, Exer. 2], these procedures are negatives of each other. We choose to use the conceptual method of non-abelian cohomology. In case F is a non-archimedean local field with residue field κ and F un denotes its maximal unramified subextension within F s (with κ the residue field of F un ) it is known from local class field theory that the natural map H 2 (F un /F, F un ) H 2 (F s /F, F s ) is an isomorphism, and the normalized valuation mapping F un Z induces an isomorphism H 2 (F un /F, F un ) H 2 (F un /F, Z) δ H 1 (Gal(F un /F ), Q/Z). There now arises the question of choice of topological generator for Gal(κ/κ): arithmetic or geometric Frobenius? We choose to follow

4 KIRSTEN EISENTRÄGER Deligne s convention and work with geometric Frobenius; in [CF86, p. 130] and [Ser79, p. 193] the arithmetic Frobenius generator is used instead. Via evaluation on the chosen topological generator, our conventions lead to a composite isomorphism inv F : Br(F ) Q/Z for non-archimedean local fields F. If one uses the arithmetic Frobenius convention, then by adopting the crossed-product algebra method to define the isomorphism Br(F ) H 2 (F s /F, F s ) one gets the same composite isomorphism inv F since the two sign differences cancel out in the composite. (Warning: In [CF86] and [Ser79] the Brauer group of F is defined to be H 2 (F s /F, F s ), and so the issue of choosing between non-abelian cohomology or crossed-product algebras does not arise in the foundational aspects of the theory; however, it implicitly arises in comparison with central simple algebras, such as in [CF86, pp. 137-8] where the details are omitted.) Since Br(R) is cyclic of order 2 and Br(C) is trivial, for archimedean local fields F there is a unique injective homomorphism inv F : Br(F ) Q/Z. By [CF86, Thm. 3, p. 131], for a finite extension F /F of nonarchimedean local fields, composition with the natural map Br(F ) Br(F ) carries inv F to [F : F ] inv F. By [Ser79, p. 194, Cor. 3], inv F (E) has order [E : F ] for any central division algebra E over F. These assertions are trivially verified to also hold for archimedean local fields F. Theorem 3.5. Let F be a global field. There is an exact sequence 0 Br(F ) P Br(F v ) Q/Z 0, v M F where M F denotes the set of nonequivalent nontrivial absolute values of F and F v denotes the completion of F at v. The first map is defined via the maps Br(F ) Br(F v ) given by extension of scalars, and the second map is given by summing the local invariants. Proof. This is [CF86, 9.7, 11.2]. For a global field F and a central division algebra E over F, write inv v (E) for inv Fv (E F v ). The theorem says that a central division algebra E over a global field F is uniquely determined up to isomorphism by its invariants inv v (E), and that these may be arbitrarily assigned subject to the condition inv v (E) = 0 for all but finitely many v and invv (E) = 0. Moreover, the order of [E] in Br(F ) is the least common denominator of the inv v (E) Q/Z.

THE THEOREM OF HONDA AND TATE 5 If K is any field then for a class c Br(K) its period is its order and its index is [ : K] with the unique central division algebra over K representing the class c. It is a classical fact that the period divides that index and that these integers have the same prime factors (see [Ser79, X.5], especially Lemma 1 and Exercise 3), but in general equality does not hold. (There are function fields of complex 3-folds for which some order-2 elements in the Brauer group cannot be represented by a quaternion algebra, and counterexamples with less interesting fields were first discovered by Brauer.) We have noted above that over local fields there is equality of period and index (the archimedean case being trivial). We need one more theorem that gives such a result over global fields: Theorem 3.6. For a central division algebra E over a global field F, the order of [E] in Br(F ) is [E : F ]. As a special (and very important) case, elements of order 2 in Br(F ) are precisely the Brauer classes of quaternion division algebras for a global field F ; as noted above, this fails for more general fields. We shall require this special case for Q and real quadratic fields in our study of k-simple A for which f A has a real root. Since Theorem 3.6 does not seem to be explicitly stated in any of the standard modern references on class field theory (though it is cryptically alluded to in [AT68, p. 105]), we give a proof below. Proof. Let E have degree n 2 over F and let d be the order of [E] in Br(F ), so d n. Note that d is the least common multiple of the local orders d v of [E F F v ] Br(F v ) for each place v of F, with d v = 1 for complex v, d v 2 for real v, and d v = 1 for all but finitely many v. Using these formal properties of the d v s, we may call upon the full power of global class field theory via Theorem 5 in [AT68, Ch. X] to infer the existence of a cyclic extension L/F of degree d such that [L v : F v ] is a multiple of d v for every v of F (here, v is any place on L over v, and the constraint on the local degree is only non-trivial for d v > 1). In the special case d = 2 (the only case we will require) one only needs weak approximation and Krasner s Lemma rather than class field theory: take L to split a separable quadratic polynomial over F that closely approximates ones that define quadratic separable extensions of F v for each v such that d v = 2. Since restriction maps on local Brauer groups induce multiplication by the local degree on the local invariants, it follows that E F L is locally split at all places of L, whence by the injectivity of the map from the global Brauer group into the direct sum of the local ones (for L) we

6 KIRSTEN EISENTRÄGER conclude that the Galois extension L/F of degree d splits E. (The existence of cyclic splitting fields for all Brauer classes is proved for number fields in Tate s article in [CF86] and is proved in general in Weil s Basic Number Theory, but neither reference seems to exert much control on the degree of the global cyclic extension.) By a general result concerning Brauer groups of arbitrary fields [Ser79, Ch. X, 5, Lemma 1], every Brauer class split by a Galois extension of degree r is represented by a central simple algebra with degree r 2. Applying this fact from algebra in our situation, [E] = [C] for a central simple algebra C of degree d 2 over F. But each Brauer class is represented by a unique central division algebra, and so C must be F -isomorphic to a matrix algebra over E. Since [C : F ] = d 2 and [E : F ] = n 2 with d n, this forces d = n as desired. 4. Statement of Theorem We will keep the notation of Section 2. Theorem 4.1. (1) Let k be a finite field of size q = p a. The map A π A defines a bijection from the set of k-isogeny classes of simple abelian varieties over k to the set W (q)/(conjugacy). Let A M(k) be simple. Let E := End 0 k(a) and let F := Q[π A ] E. Then E is a division algebra whose center is F. (2) The division algebra E does not split at any real place of F, and E splits at all finite places of F not dividing p = char(k). For a place v of F dividing p we have (4.1) inv v (E) = v(π A) v(q) [F v : Q p ] mod 1, where F v denotes the completion of F at v. We have (4.2) 2 dim A = [E : F ] 1/2 [F : Q]. The injectivity of the map A π A was proved by Tate, and the surjectivity of this map was proved by Honda [Hon68]. Part(2) of the theorem was proved by Tate in [Tat66]. An interesting consequence of (4.2) is that if K E is a maximal commutative subfield over F (so [K : F ] = [E : F ]) then [K : Q] = 2dimA. Hence, every k-simple abelian variety over a finite field k is of CM type over k. 5. Examples Let π W (q) C be a Weil q-integer in C. Grant Theorem 4.1.

THE THEOREM OF HONDA AND TATE 7 5.1. Real Case: Assume F = Q[π] has a real embedding, and fix φ such an embedding Q[π] R C. Then φ(π) = φ(π). In this case φ(π 2 ) = φ(π) φ(π) = q = p a, so π 2 = p a. Of course, ± q is a Weil q- integer for any q, so by Theorem 4.1(1) it does arise from a k-simple A that is unique up to k-isogeny. There are two possible cases to consider. (1) When a is even, we have F = Q. There is only one real place v, and for this place we have inv v (E) = 1/2, which implies that E must be ramified at p as well with inv p (E) = 1/2. This implies that the order of E in Br(Q) is 2, and so (Theorem 3.6) E is a quaternion division algebra over Q which is ramified at and p. The formula 2 dim A = [E : F ] 1/2 [F : Q] implies that dim A = 1. Hence, the characteristic polynomial of f A Z[T ] with a factor T p a/2 has degree 2 and constant term q = p a, so it must be (T p a/2 ) 2. Since a p (A) = 2p a/2 pz, A is a supersingular elliptic curve. For any abelian variety X over any field K, the faithfulness of V l (X) as a Q l End 0 K(X)-module for any prime l char(k) ensures that dim Q End 0 K(X) (2dimX) 2. Thus, for Q-dimension reasons E = End 0 k(a) = End k (A) Q, i.e. all k-endomorphisms are already defined over k. Note that the surjectivity in Theorem 4.1(1) ensures conversely that this case really does occur over any k! Conversely, we know from the theory of supersingular elliptic curves that if A is a supersingular elliptic curve over k whose k-endomorphisms are defined over k then E = End 0 k(a) is a quaternion division algebra over Q, so the central subfield F = Q(π) E must be Q. (2) When a is odd, we have F = Q(π) = Q(p 1/2 ). In F we have two real places, and the invariant of E at those two places is 1/2 by the theorem. Since there is only one place v of F dividing p, this means that E is not ramified at v p. The division algebra E has order 2 in Br(F ), so [E : F ] = 4 by Theorem 3.6. By formula (4.2), dim A = 2. The characteristic polynomial f A of π A in Z[T ] is therefore a quartic with ±p a/2 as a root. By Theorem 1.1, f A is a power of an irreducible polynomial in Q[T ] since A is k-simple, so f A = (T 2 p a ) 2. Thus, the characteristic polynomial of the q 2 -Frobenius for A over a degree 2 extension k of k is (T p a ) 4. By the existence aspect in the real case (1) applied over the field k with size p 2a, there exists a supersingular elliptic curve A 0 over k with characteristic polynomial (T p a ) 2. By Theorem 1.1 all choices of A 0 are k -isogenous, and in (1) we saw that such A 0 are precisely the supersingular

8 KIRSTEN EISENTRÄGER elliptic curves over k whose geometric endomorphisms are defined over k. The characteristic polynomial of the q 2 -Frobenius of A 0 A 0 is (T p a ) 4, so A /k is k -isogenous to A 0 A 0. Thus, this is the case of an abelian surface that is k-simple but is isogenous over k to A 0 A 0 for some (equivalently, any) supersingular elliptic curve A 0 over k whose geometric endomorphisms are all defined over k. Observe that necessarily A is k -isogenous to Res k /k(a 0 ). By the surjectivity aspect of Theorem 4.1(1) such a case must always occur. (That is, this restriction of scalars must be k-simple for any finite field k of size p a with odd a and the quadratic extension k /k.) 5.2. General case: F = Q(π) is totally imaginary. For any embedding φ : Q[π] C we have φ(π)φ(π) = q, so φ(π) = q/φ(π) = φ(q/π). Thus, φ(f ) C is stable under complex conjugation with φ 1 (φ(x)) independent of φ since this is true for x = π. Then F is a CM-field, i.e. a quadratic totally imaginary extension of a totally real field F 0. Let x = φ 1 (φ(x)) for any x F, so π = q/π. Thus, F 0 = Q(π + π) is the totally real subfield of F. The division algebra E splits at each place v which does not divide p. (There are no real places.) For a place v dividing p, in Q/Z we have (5.1) (5.2) inv v (E) + inv v (E) = 0, if v v, inv v (E) = 0, if v = v. Formula (5.1) follows from ππ = q, v(π)+v(π) = v(q), and the formula for the invariant in (4.1) since [F v : Q p ] = [F v : Q p ]. Equation (5.2) follows from formula (4.1) of the theorem and the fact that [F v : Q p ] is even when v = v. If k is the prime field, k = F p, then E is split everywhere since v(π)/v(p) is an integral multiple of 1/e v and e v [F v : Q p ] (so for any prime v dividing p, formula (4.1) of the theorem shows that inv v (E) = 0 in Q/Z). Thus, E = F and hence End k (A) is commutative (recall k = F p ). 5.3. The remaining possibilities when dim A = 1. If dim A = 1, then 2 = [E : F ] 1/2 [F : Q]. The case F = Q has been settled, as has the case of real quadratic F (with A necessarily an abelian surface). Thus, if A is either an ordinary elliptic curve or a supersingular elliptic curve whose geometric endomorphism ring (a quaternion division algebra over Q ramified at precisely p and ) is strictly larger than E then E = F is an imaginary quadratic field. By Theorem 4.1 this exhausts

THE THEOREM OF HONDA AND TATE 9 all cases with E = F and F imaginary quadratic. Since F = Q(π), we conclude (via 5.1) that these are precisely the elliptic curves over finite fields of size q such that the q-frobenius is not multiplication by a rational integer. 6. Injectivity of the map A π A Injectivity proof. Let A and B be two k-simple abelian varieties such that π A = π B in W (q) modulo conjugacy. Let f A and f B be the characteristic polynomials of π A and π B, respectively. Since A is simple, f A is a power of an irreducible polynomial h Q[T ], say f A = h m with m 1. Since π A = π B and B is simple, f B = h n with n 1. By relabeling we may assume m n, so f B divides f A. Then by Theorem 1.1 we see that B is isogenous to an abelian subvariety of A over k. Since A and B are k-simple, this inclusion must be an isogeny. 7. Surjectivity of the map A π A To prove surjectivity we have to construct sufficiently many simple abelian varieties over k to exhaust all the conjugacy classes of Weil q-integers. But we cannot easily write down abelian varieties of dimension greater than 2 over k, so how to proceed? We will construct abelian varieties with complex multiplication over C, realize them over number fields that we embed in finite extensions of Q p, and then reduce them to obtain abelian varieties over finite fields. Notation: We will call a Weil q-integer π effective if π is conjugate to the q-frobenius of a simple abelian variety over k. (That is, there exists a k-simple abelian variety A and an embedding Q[π] End 0 k(a) carrying π to π A.) So our goal is to prove that all Weil q-integers are effective. In the definition of effectivity there is no loss of generality in allowing A to be k-isotypic (i.e., isogenous to a power of a k-simple abelian variety) rather than k-simple, since such an A has endomorphism algebra that is a matrix algebra over a division algebra and so its central Frobenius endomorphism φ generates a commutative central subfield that preserves any k-simple abelian subvariety and on which φ is its Frobenius endomorphism. A basic example of an isotypic abelian variety over a field is a CM abelian variety (i.e., an abelian variety whose endomorphism algebra contains a commutative subfield with degree over Q equal to twice the dimension of the abelian variety). We will first prove the following lemma. Lemma 7.1. Let N be an integer, N 1. If π N is effective, then π is effective as well.

10 KIRSTEN EISENTRÄGER Proof. Let k be an extension of k of degree N. Let A be a simple abelian variety over k such that π N is conjugate to π A. Let A be the abelian variety over k which is the restriction of scalars of A. Let l be a prime, l p. From the functorial characterization of Weil restriction of scalars, as Q l [G k ]-modules (with G k = Gal(k/k) for a separable closure k/k ) we have V l A = Ind G k G k V la. From this one can see that the characteristic polynomial f A of π A and the characteristic polynomial f A of π A are related via f A (T ) = f A (T N ) (Lemma 9.1 in the Appendix). If A i is the isogeny decomposition of A into k-simple factors, then π is a root of f A = f Ai, so π is a root of some f Ai, say i = 1 without loss of generality. Since A 1 is k-simple, f A1 is a power of an irreducible polynomial h over Q and the Frobenius endomorphism π A1 satisfies h(π A1 ) = 0 on A 1, or more specifically the Q-subalgebra Q[π A1 ] End 0 k(a 1 ) is abstractly isomorphic to Q[x]/(h(x)) via x π A1. Thus, we can define an embedding Q[π] End 0 k(a 1 ) with π π A1. Let π be a Weil q-integer. In the following let E be the division algebra obtained from π and the information in part (2) of Theorem 4.1; i.e., E is the central division algebra over F = Q(π) whose local invariants are as in the theorem and so are completely determined by π. (Explicitly, inv v (E) = 1/2 for real places v of F, if any exist, and for non-archimedean v the fraction v(π)[f v : Q p ]/v(q) represents inv v (E) Q/Z; this vanishes for such v not over p since π is a Weil q-integer.) By Theorem 3.5, such an E indeed exists and is unique up to isomorphism. Lemma 7.2. There exists a CM field L containing F = Q(π) such that L splits E and such that [L : F ] = [E : F ] 1/2. Proof. Since π is a Weil q-integer, F is a totally real or CM field. In the totally real case, as in 5.1 we must have F = Q or F = Q( p) with [E : F ] 1/2 = 2, so L = F ( p) works. If F is CM with maximal totally real subfield F 0, then we take L = F F0 L 0 with L 0 /F 0 a totally real extension of degree [E : F ] 1/2 such that for all places v 0 of F 0 over p we have two properties: (i) there is exactly one place w 0 of L 0 over v 0, and (ii) w 0 is unramified (resp. totally ramified) over v 0 if the quadratic extension F/F 0 is ramified (resp. unramified) over v 0. Such an L 0 can be found by weak approximation, and by construction for each place w of L over p with restrictions to L 0, F, and F 0 denoted w 0, v, and v 0,

we have THE THEOREM OF HONDA AND TATE 11 [L w : F v ] = [L 0,w0 : F 0,v0 ] = [L 0 : F 0 ] = [E : F ] 1/2. Hence, L F E is everywhere locally split over L, so it is split. Definition 7.3. Let A be an abelian variety over a field K. Let L be a number field. We say that A is of type (L) if there is a ring map i : L End 0 K(A) such that [L : Q] = 2 dim A. Lemma 7.4. Let L be a CM field containing F = Q(π) and such that L splits E. Then there exists an abelian variety A of type (L) over a finite extension of Q p such that A has good reduction and such that the Frobenius of its reduction is a power π N of π. Note that in this lemma there is no up to Galois conjugacy ambiguity; we are saying that the entire CM field L acts on A (in the isogeny category over a p-adic field) such that the element π N in its subfield F is the Frobenius endomorphism of the reduction of A. We will prove Lemma 7.4 in the next subsection. The three lemmas together clearly show that any given Weil q-integer π is effective. 7.1. Proof of Lemma 7.4. Let L be a CM field with [L : Q] = 2 dim A. Let ρ be the automorphism of L of order 2 which is induced by complex conjugation for each embedding L C. Intrinsically, ρ is the unique nontrivial automorphism of L over its maximal totally real subfield. Let C be an algebraically closed field of characteristic 0. Let Φ be a CM type for L, i.e. Φ is a subset of Hom Q alg (L, C) such that (7.1) (7.2) Φ Φρ = and Φ Φρ = Hom(L, C). Definition 7.5. An abelian variety A over a subfield F C is of type (L, Φ), if A is of type L over F and the action of C Q L on the tangent space t AC is through the quotient φ Φ C φ (where L acts on C φ = C through φ). Lemma 7.6. There exists an abelian scheme of type (L, Φ) defined over the ring of integers of a number field contained in C. Proof. This is proved in the notes for Tong Liu s talks in the seminar. From now on let us assume that C is an algebraic closure of Q p. For each place w of L such that w p, let L w be the completion of L with respect to w.

12 KIRSTEN EISENTRÄGER We identify Hom Qp (L w, C) with its image in Hom(L, C) and denote this image by H w (so Card(H w ) = [L w : Q p ]). Let Φ w := Φ H w. We have a decomposition (7.3) Q p L = w p L w and the disjoint unions (7.4) Hom(L, C) = w p H w (7.5) Φ = w p Φ w. Lemma 7.7. Let A be an abelian scheme of type (L, Φ) defined over the ring of integers O of a finite extension of Q p. Let k 0 be the residue field of O, and let q 0 = Card(k 0 ). Let A 0 be the reduction of A modulo the maximal ideal of O. Then there exists an element π 0 L such that i(π 0 ) End(A) induces the Frobenius π A0 End 0 k(a 0 ), and we have (7.6) for each w p. w(π 0 ) w(q 0 ) = Card(Φ w) Card(H w ) = Card(Φ w) [L w : Q p ] Proof. This was proved by Shimura and Taniyama and will be presented in the seminar by Brian Conrad. Now we can prove Lemma 7.4. Proof of Lemma 7.4. Let π, q, F, E as before, and assume that L satisfies the hypotheses of Lemma 7.4. Let z z ρ be the complex conjugation on L. We will now show how we can choose the set Φ such that we have w(π) (7.7) w(q) = Card(Φ w) for each place w of L with w p. Card(H w ) Let w be one such place, and let v be the place of F such that w v. Let us put n w = w(π) w(q) Card(H w) = w(π) w(q) [L w : Q p ] = v(π) v(q) [L w : F v ] [F v : Q p ]. By functoriality of the local invariants and the hypothesis on inv v (E), inv w (L F E) = [L w : F v ] v(π) [F v(q) v : Q p ] = n w in Q/Z. The fact that L splits E implies that n w is an integer for each w. We also clearly have (7.8) n w 0,

THE THEOREM OF HONDA AND TATE 13 and since ππ ρ = q we can conclude (7.9) n w + n ρw = Card H w = Card H ρw. We will now choose the CM type Φ by choosing suitable sets Φ w and then letting Φ := w Φ w. Given integers n w satisfying conditions (7.8) and (7.9) we can see that we can choose subsets Φ w H w such that their union Φ := w Φ w satisfies conditions (7.1) and (7.2) and n w = Card Φ w. Let us make such a choice of Φ. Then condition (7.7), namely w(π) w(q) = Card(Φ w) Card(H w ) is satisfied by construction. By Lemma 7.6 there exists an abelian scheme of type (L, Φ) defined over the ring of integers O of a finite extension of Q p, and by Lemma 7.7 there exists π 0 L such that (7.10) w(π) w(q) = Card(Φ w) Card(H w ) = w(π 0) w(q 0 ). Here π 0 is a Weil q 0 -integer, with π 0 L, which is conjugate to the Frobenius of the reduction A 0 of A. Now we are almost finished. If we replace the ring O in Lemma 7.7 by an unramified extension of degree N 0 we replace π 0 by π N 0 0. We can now finish the proof of Lemma 7.4 with the following lemma. Lemma 7.8. Let π be a Weil q-integer in L and π 0 a Weil q 0 -integer in L which satisfy w(π) = w(π 0) w(q) w(q 0 for all w. Then there exist positive ) integers N and N 0 such that π N = π N 0 Proof. By replacing π and π 0 by powers we may assume that q = q 0. Then w(π) = w(π 0 ) for each place w of L such that w p. At the other finite places π and π 0 are units, because they divide a power of p. At the infinite places π and π 0 have the same absolute value, namely q 1/2. Hence π/π 0 has absolute value 1 at each place of L, which implies that π/π 0 is a root of unity. Hence some power of π/π 0 is equal to 1, and the lemma follows. 0. 8. Proof of part 2 of Theorem 4.1 In this section we will prove the end of Theorem 4.1(1) and Theorem 4.1(2). 8.1. Statement of result and preliminary arguments. Let A be a simple abelian variety over a finite field k with size q. Let π A be the q-frobenius of A. Let E := End 0 k(a). Then E is a division algebra. Let us recall Theorem 4.1(2) and the end of Theorem 4.1(1):

14 KIRSTEN EISENTRÄGER Theorem 8.1. (1) The center of E is F = Q(π). (2) E splits at every finite place v of F such that v does not divide p. (3) E does not split at any real place. (4) The invariant inv v (E) for a place v p is given by the formula (8.1) inv v (E) = v(π A) v(q) [F v : Q p ] mod 1. (5) We have 2 dim A = [E : F ] 1/2 [F : Q]. Formula (8.1) will be proved separately. proved now. Everything else will be Proof. Part (1): We know that E is a division algebra, so its center is a field. Let G k := Gal(k/k). For a prime l p let T l A be the l-adic Tate module and let V l (A) := T l A Q l. Let F l := Q l F, and let E l := E Q l. Let S := End V l (A), and T := F l. From Tate s isogeny theorem we have E Q l = EndGk (V l A), so the centralizer of T in S is E l because G k is topologically generated by the q-frobenius. Since E l is semisimple, we can apply the double centralizer theorem and conclude that the centralizer of E l in End V l (A) is F l. So F l is the center of E l and hence F is the center of E. Part (5): If A is a k-simple abelian variety, then the characteristic polynomial f A of π A is a power of a Q-irreducible monic polynomial h A. Then deg h A = [F : Q]. Say f A = h e A. This implies that 2 dim A = deg f A = e deg h A = e [F : Q]. We also have [E : Q] = e 2 [Q(π A ) : Q]. This comes from the dimension computation in the proof that E Q Q l = EndGk (V l A) in [Tat66]. (Alternatively, see [Mum70, Thm. 2(a), Appendix I] for a proof.) Part (2): Since f A = h e A with h A irreducible over Q and since V l A is a semisimple F Q l -module, for a prime l different from p, the Tate module is a free module of rank e over F l = F Q l. By Tate s theorem, we have F l F E = Q l Q E = E l is isomorphic to End Fl V l (A), that is, to the algebra of e e matrices over F l. Since F l = v l F v it follows that E v = F v Q E is a matrix algebra over F v for v l. That means E splits at all primes v l. Part (3): Let A be a simple abelian variety over k. Assume that F = Q(π) has a real place v. We may then view π as a real number, which has absolute value q 1/2, where q = Card(k). Consequently π is a root of the polynomial T 2 q.

THE THEOREM OF HONDA AND TATE 15 If q is a square, then π is rational, F = Q, and since we already proved that the formula 2 dim A = [E : F ] 1/2 [F : Q] holds, we can rule out the possibility that E = F = Q. Hence E is nonsplit. We also proved that F is the center of E, and that E splits at every finite place v of F that does not divide p. By Theorem 3.5, E must be non-split at p and, and the invariant must be 1/2 at those two places. This implies [E : F ] = 4, so E is a quaternion division algebra which does not split at p and the real place v =. That means that the reasoning in 5.1 applies, and so A is a supersingular elliptic curve over k whose k-endomorphisms are all defined over k (and necessarily f A = (T q 1/2 ) 2 ). If q is not a square in Q, then F = Q( p) is a real quadratic field. Also, since f A must be a power of an irreducible polynomial over Q, we must have f A = (T 2 q) e for some e 1. Over a quadratic extension k /k, A = A /k therefore has associated q 2 -Frobenius polynomial f A = (T q) 2e, so by the case just treated we see that A is k -isogenous to A 0 A 0 for a supersingular elliptic curve A 0 over k whose geometric endomorphisms are all defined over k. By the previous case, the k -endomorphism algebra of A 0 is the quaternion division algebra D p over Q that is ramified at exactly p and. Let E be the k -endomorphism ring of A. Then E is a 2 2 matrix algebra over D p. Since F = Q[π], E is the centralizer of F in E. By the remark after Theorem 9.2 in the appendix we have that E E Q F D p Q F in Br(F ). Since D p does not split over R it follows that E does not split at either of the real primes of F. The rest of 8 is devoted to the computation of inv v (E) for the case v p. We will follow the method of Tate as explained in [MW71], except that we cut down on the input from non-commutative ring theory. 8.2. Dieudonné modules. To compute inv v (E) for v p we have to replace the l-adic Tate module by a suitable object. Notation: Let k be a perfect field of characteristic p > 0. Let W be the Witt vectors of k; e.g., if k is finite of size q = p a then W is the ring of integers in the unramified extension of Q p of degree a. Let σ be the unique automorphism of W which reduces to the map x x p on the residue field k. Let D k := W [F, V], where F, V are indeterminates subject to the relations

16 KIRSTEN EISENTRÄGER (1) FV = VF = p (2) Fα = α σ F and αv = Vα σ for α W. Elements of the Dieudonné ring D k have unique expressions as finite sums a 0 + a j F j + b j V j j>0 j>0 with coefficients in W, so the center of D k is clearly Z p [F a, V a ] if k has finite size p a and it is Z p otherwise (i.e., if k is infinite). Some of the main conclusions in classical Dieudonné theory, as developed from scratch in [Fon77, Ch. I III], are summarized in the following theorem: Theorem 8.2. There is an additive anti-equivalence of categories G D(G) from the category of finite commutative k-group schemes of p- power order to the category of left D k -modules of finite length (or equivalently, of finite W -length). Moreover, (1) A group scheme G has order p l W (D(G)). (2) If k k is an extension of perfect fields with associated extension W W of Witt rings (e.g., the absolute Frobenius automorphism of k) then the functor W W ( ) on Dieudonné modules is naturally identified with the base-change functor on finite commutative group schemes. In particular, D(G (p) ) σ (D(G)) as W -modules. (3) Let F G : G G (p) be the relative Frobenius morphism. The σ-semilinear action on D(G) induced by D(F G ) with respect to the isomorphism D(G (p) ) σ (D(G)) equals the action of F, and G is connected if and only if F is nilpotent on D(G). Let A be an abelian variety over k with dimension g 1. Let A[p n ] be the finite commutative p n -torsion group scheme of rank (p n ) 2g. We associate with A the p-divisible group A[p ] of height 2g. Rather generally, for any p-divisible group G = (G n ) n 1 over k with height h 1, we let D(G) denote the D k -module lim D(G n ). Then by the same style of arguments used to work out the Z l -module structure of Tate modules of abelian varieties in characteristic l (resting on knowledge of the size of the l-power torsion subgroups of geometric points), we use W -length to replace counting to infer that D(G) is a free right W -module of rank h with D(G)/p r D(G) D(G r ) for all r 1. The p-divisible group G is connected if and only if F is topologically nilpotent on D(G) (since this is equivalent to nilpotence of F on each D(G r )).

THE THEOREM OF HONDA AND TATE 17 In analogy with the l-adic case we will now write T p G for D(G) and T p A for the case G = A[p ]. The D k -module T p G will be the replacement for the l-adic Tate module in the classical case, even though it is contravariant in A; its D k -action is the analogue of the Galois action on l-adic Tate modules, though this Dieudonné structure remains non-trivial when k is algebraically closed (whereas the Galois action on Tate modules is trivial for such k). More Notation: Let L be the fraction field of W. For any p- divisible group G over k, let V p G := T p G Zp Q p (and write V p A for G = A[p ]). Then V p G is an L-module of rank equal to the height h of G, and it also has a left module structure over the Laurent polynomial ring D k [1/p] = L[F, 1/F] that is non-commutative if k F p. Here is the analogue of Tate s isogeny theorem for V p A: Theorem 8.3. For any abelian variety A over k, with p-divisible group denoted A[p ], the Q p -algebra map Q p Q End 0 k(a) End 0 k(a[p ]) is an injection, and if k is finite then it is an isomorphism. Proof. By Theorem 8.2 this is Q p Zp ( ) applied to the map Z p End k (A) End k (A[p ]) and this latter map is proved to be injective exactly as in the l-adic case for l p, by working with (contravariant) Dieudonné modules instead of the l-adic Tate modules. As for surjectivity for finite k, we first note that the q-frobenius endomorphism π acting W (k)-linearly on D(A[p ]) has the same characteristic polynomial (in Z[T ]) as it does as a Z l -linear endomorphism of the l-adic Tate module of A for any l p; this is true more generally for any element of End k (A), and its proof is given by using Dieudonné modules in the role of Tate modules in the proof of [Mum70, Thm. 4, p. 180]. Hence, we may copy Tate s proof of surjectivity essentially verbatim (say, as in Appendix 1 of [Mum70]) by replacing l-adic Tate modules with Dieudonné modules throughout, except that [Mum70, Lemma 6, App. 1] only applies to compute the L-dimension of the centralizer of the semisimple L-linear endomorphism F a = V p (π) acting on the L-vector space V p (A). (That is, we can compute dim L End L[F a ](V p (A)).) In order to conclude the proof, it is necessary and sufficient to prove that this L-dimension is equal to the Q p -dimension of End 0 k(a[p ]) = End Dk [1/p](V p (A)) opp. The central subalgebra F = Q[π] in the finite-dimensional semisimple Q-algebra End 0 k(a) must be semisimple and hence a finite product of fields; we do not assume A to be k-simple, so in particular F may

18 KIRSTEN EISENTRÄGER not be a field. Consider the decomposition of A and A[p ] (in the isogeny sense) according to the idempotents of F. The action by F on V p (A) (through its action on A in the isogeny category of abelian varieties over k) commutes with the action by D k [1/p] on V p (A), and F a D k [1/p] acts as V p (π) on V p (A). Since there are no nonzero k- maps among the factor abelian varieties or p-divisible groups over k on which the central subalgebra F acts through distinct quotient fields, we may reduce the proof of Tate s theorem to the case when F is a field. Let m Z[T ] be the common characteristic polynomial for V l (π) on the Q l -vector spaces V l (A) (for all l p) and for V p (π) on the L-vector space V p (A), so by the faithfulness of V p (A) as a Q p Q F -module we see that rad(m) is the minimal polynomial of π F over Q. Hence, Q p Q F v p F v with F v Q p [T ]/(m v ) for the monic irreducible factors m v Z p [T ] of m over Q p. Each m v (0) divides m(0) = q dim A, so all m v s have nonzero constant term. Since Q p Q F acts D k [1/p]-linearly on V p (A), we get a decomposition of D k [1/p]-modules V p (A) v p V p(g v ) where v p G v is the isogeny decomposition of A[p ] with respect to the idempotents of Q p Q F. Since the central element m v (F a ) D k acts on G v through the element m v (π) = 0 in F v, V p (G v ) is a left module over the quotient algebra C v = D k [1/p]/D k [1/p]m v (F a ). Using the compatible decompositions (as L-algebras and Q p -algebras respectively) End L[F a ](V p (A)) v p End L[F a ](V p (G v )) and End Dk [1/p](V p (A)) v p End Cv (V p (G v )), we are reduced to proving dim L End L[F a ](M v )? = dim Qp End Cv (M v ) for any v p and any left C v -module M v with finite Q p -dimension (e.g., M v = V p (G v )). This general equality is valid when C v is replaced with D k [1/p]/D k [1/p]h(F a ) for any h Q p [T ] that is a monic irreducible with nonzero constant term; a proof will be given in Corollary 8.7 as a consequence of some general calculations in non-commutative algebra (having no logical dependence on abelian varieties or p-divisible groups).

THE THEOREM OF HONDA AND TATE 19 8.3. Restatement via left D k [1/p]-modules. Take k to be finite of size q, and A a k-simple abelian variety over k with endomorphism algebra E, so E is a central simple algebra over the subfield F = Q[π]. Under the decomposition E Q p v p E v with E v = E F F v, we get a corresponding decomposition of A[p ] in the isogeny category of p-divisible groups over k, as a product of nonzero p-divisible groups G v on which F Q p acts through its quotients F v (here we use the elementary injectivity aspect of Theorem 8.3 to know that each G v is nonzero). Since the Dieudonné functor is fully faithful and contravariant, and the map in Theorem 8.3 is an isomorphism for finite k, we may identify the central F Q Q p -algebra E Q p with the opposite F Q Q p -algebra to End Dk [1/p](V p A), where F = Q(π) acts through functoriality via its action on A in the isogeny category over k. In particular, V p (π) is the action of the central element F a D k [1/p], and so we get E v End 0 k(g v ) End Dk [1/p](V p G v ) opp. as finite-dimensional F v -algebras. We conclude that the right side is a central simple F v -algebra, and our problem is to compute its invariant. 8.4. Passage to a central simple F v -algebra quotient of D k [1/p]. We continue with the notation and hypotheses as in 8.3. Since π in the ring of integers of F acts on A as the q-frobenius over k, on the finite-dimensional Q p -vector space V p (G v ) the Q p -linear operator F a = V p (π) acts with a characteristic polynomial that divides the common characteristic polynomial m Q[T ] Q p [T ] for the q-frobenius on each V l (A) over Q l (for l p) and V p (A) over L. (Explicitly, for all n Z the integer m(n) is the degree of n π acting on A.) As we noted in the proof of Theorem 8.3, m is a power of the minimal polynomial of π F over Q, and so the decomposition F Q Q p v p F v corresponds to the pairwise distinct monic irreducible factors m v of m over Q p. That is, F v = Q p (π) with π F v having minimal polynomial m v over Q p. In particular, m v (0) divides m(0) = q dima, so all m v s have nonzero constant term. For each v p, the F v -action on the nonzero V p (G v ) commutes with the D k [1/p]-action (as it arises from an action of F v on the p-divisible group G v in the isogeny category over k), and the central element m v (F a ) D k [1/p] acts as multiplication by the element m v (π) F v that is zero. In other words, V p (G v ) is a nonzero module over the ring Recall that q = p a. C v := D k [1/p]/D k [1/p]m v (F a ).

20 KIRSTEN EISENTRÄGER Theorem 8.4. Let h Q p [T ] be a monic irreducible with h T, and let K = Q p [T ]/(h). The central K-algebra C = D k [1/p]/D k [1/p]h(F a ) (with T acting as F a ) is central simple. In order to prove Theorem 8.4, we require some preparations. Recall that L denotes the fraction field of W (k). We have dim L C = deg(h(t a )) = a[k : Q p ] due to: Lemma 8.5. Let λ L[T ] be monic with positive degree d and nonvanishing constant term. Every g D k [1/p] can be uniquely written g = h λ(f) + (c 0 + c 1 F + + c d 1 F d 1 ) with c 0,..., c d 1 L and h D k [1/p]. In particular, the left D k [1/p]- module D k [1/p]/D k [1/p] λ(f) has dimension d as a left L-vector space. Proof. The uniqueness is easily proved by chasing least-degree coefficients (when things are nonzero). For existence it suffices to treat the cases g = F r with r d and r < 0. These go as in the commutative case (for r < 0 we use the non-vanishing of the constant term of λ). Since C is an L Qp K-module with finite Q p -dimension, we obtain dim K C = a[l : Q p ]. Let (L Qp K)[F] denote the K-algebra defined by the relations F a = 1 π (for π K) and F x = (σ 1)(x) F for x L Qp K, with σ Gal(L/Q p ) the absolute Frobenius. This is easily checked to be a central K-algebra with dimension a[l : Q p ] = a 2. There is a unique map of central K-algebras (L Qp K)[F] C sending F to F and L to L by the identity map, and this is clearly surjective. By consideration of K-dimensions, it is an isomorphism. Theorem 8.4 therefore will follow from: Theorem 8.6. The central K-algebra (L Qp K)[F] is K-isomorphic to a matrix algebra over a cyclic K-algebra. Remark. Cyclic algebras are a special class of central simple algebras, studied by Dickson before the advent of the general theory. Proof. We first construct the cyclic K-algebra over which the K- algebra (L Qp K)[F] will be proved to be a matrix algebra. Let f be the absolute residue degree of K, and let g := gcd(f, a). Let L K K denote the maximal unramified subextension of K that embeds into L (over Q p ), so g = [L K : Q p ]. We arbitrarily choose one of the g embeddings L K L as cyclic extensions of Q p, and let LK denote the resulting linearly disjoint compositum L L K K.

THE THEOREM OF HONDA AND TATE 21 Clearly LK/K is unramified with degree a/g. Let θ : L Qp K LK be the resulting projection map. Let be the central K-algebra (LK)[F ] defined with relations (F ) a/g = π K and F x = σ (x ) F where σ : LK LK is the L K-automorphism of order a/g induced by σ g on L and the identity on K. Thus, σ (θ(x)) = θ((σ g 1)(x)) for x L Qp K, and we have a natural isomorphism of K-algebras g L Qp K LK defined by j=1 x (θ(x), θ((σ 1)(x)),..., θ((σ g 1 1)(x))). The central K-algebra is a cyclic K-algebra; in classical notation, = (LK/K, σ, π), where LK is a cyclic (unramified) extension of K and σ is a chosen generator of its Galois group. Now consider the natural map of central K-algebras ξ : (L Qp K)[F] M g ( ) defined by θ(x) 0... 0 0 θ((σ 1)(x)) 0 0 x...... 0 0... θ((σ g 1 1)(x)) for x L Qp K and 0 1 0... 0 0 0 1... 0 F. 0......... 0... 1 F 0 0... 0 The map ξ is obviously a well-defined K-algebra homomorphism, and it is easy to check that both sides have the same Q p -dimension. To prove that ξ is an isomorphism, it suffices to check surjectivity. The decomposition of L Qp K as a product of copies of LK shows that the image of ξ contains the diagonal matrices whose entries are elements of LK. Moreover, ξ(f g ) is the diagonal matrix whose diagonal entries are all equal to F. Thus, by the definition of we see that the image of ξ contains all diagonal matrices in M g ( ).

22 KIRSTEN EISENTRÄGER Left-multiplication by ξ(f) carries the set of diagonal matrices onto the set of matrices of the form 0 δ 2 0... 0 0 0 δ 3... 0.. 0........ 0... δ g F δ 1 0 0... 0 with arbitrary δ 1,..., δ g, and so since F has a multiplicative inverse (as (F ) a/g = π K ) it follows that the image of ξ contains the matrix 0 1 0... 0 0 0 1... 0.. 0......... 0... 1 1 0 0... 0 Powers of this matrix, together with arbitrary diagonal matrices in M g ( ), generate M g ( ) under the operations of multiplication and addition. This establishes surjectivity. Remark. For later purposes, it is convenient to describe the cyclic algebra in terms of an arithmetic Frobenius generator φ of the Galois group of the unramified extension LK/K. With notation as used above, we claim that = (LK/K, σ, π) is K-isomorphic to the cyclic K-algebra (LK/K, φ, π f/g ). This amounts to checking that σ f/g = φ in Gal(LK/K), and so we just have to compare these on the residue field of L (viewed inside that of LK). Since σ reduces to x x pg on the residue field of L, σ f/g reduces to x x pf on this residue field. But f is the residual degree for K over Q p, so we are done. It is an elementary result in the theory of simple rings that (up to isomorphism) there is a unique simple module over such a ring and that all finite modules are isomorphic to a direct sum of copies of this simple module. Thus, up to isomorphism there is a unique simple D k [1/p]- module V killed by the central element h(f a ), and so for any nonzero left C-module M with finite Q p -dimension we have M V r for some r 1 (with C denoting the quotient of D k [1/p] modulo the central element h(f a )). In particular, End Dk [1/p](M) is a matrix algebra over the division ring D = End C (V ). Since there is an isomorphism of left C-modules C V r 0 for some r 0 1, so M r0 (D) = End C (C) = C opp, D has center K and in the Brauer group of K we have D C opp.

THE THEOREM OF HONDA AND TATE 23 Hence, the class of End Dk [1/p](M) opp in Br(K) is the same as that of C for any nonzero finitely generated left C-module M. For any such M we have the following result that completes the final step in the proof of Theorem 8.3: Corollary 8.7. Let h Q p [T ] be a monic irreducible with h T, and let C = D k [1/p]/D k [1/p]h(F a ). For any finite left C-module M, (8.2) dim L End L[F a ](M) = dim Qp End C (M). Since L[F, 1/F] = D k [1/p] and End C (M) = End L[F] (M) (F acts invertibly on M, as h(0) Q p ), the equality [L : Q p ] = a suggests the possibility that one may be able to deduce (8.2) by some general nonsense with Galois descent. However, it seems that the proof requires fine structural information concerning C. Proof. Let V be a simple left C-module and let K = Q p [T ]/(h). We have M V r for some r 0. The two sides of (8.2) for M are each r 2 times the two sides of (8.2) for V in the role of M. Hence, the truth of (8.2) is independent of the choice of r, so it suffices to verify this identity for one M 0. We choose M = C. Since End C (C) = C opp (with action through right multiplications on C), dim Qp End C (C) = [L : Q p ] dim L (C opp ) = [L : Q p ] deg h(t a ) by Lemma 8.5 (with λ = h(t a ) satisfying λ(0) 0 since the irreducible monic h is not T ). Because deg h = [K : Q p ] and [L : Q p ] = a we may rewrite this as dim Qp End C (C) = [L Q p K : Q p ]a 2, [L : Q p ] and so by expressing an L-dimension as [L : Q p ] 1 times a Q p -dimension it is equivalent to prove dim Qp End L[F a ](C) = [L Qp K : Q p ]a 2. It therefore suffices to show that End L[F a ](C) is isomorphic as a Q p - algebra to an a a matrix algebra over L Qp K. Let t K = Q p [T ]/(h) denote the residue class of T, so K = Q p [t]. We saw above Theorem 8.6 that as L[F a ]-algebras C (L Qp K)[F] where the right side has relations F a = 1 t and F x = (σ 1)(x) for all x L Qp K. Hence, the L[F a ]-endomorphism ring of C is the same as the L Qp K-endomorphism ring of C, and this in turn is indeed an a a matrix algebra over L Qp K.